One way to examine the appropriateness of QMEs is to study their thermodynamic behavior [1]}, and especially their steady states. Here, we focus on the equilibrium steady state of QMEs when the system is coupled with a single thermal bath. It is well known [2]} that the Lindblad equation obtained from secular approximation results in the Gibbs state of the system Hamiltonian as a steady state. However, if we accept that the total system, consisting of the system of interest and the bath, thermalizes, for example by satisfying the condition required for the eigenstate thermalization hypothesis [3]}, [4]}, [5]}, [6]}, the steady state of the reduced density operator will be the so-called mean force Gibbs (MFG) state [7]}, [8]}, [9]}, [10]}, which is obtained by tracing the Gibbs state for the total Hamiltonian over the bath variables [11]}, [12]}, [13]}. Therefore, the Lindblad equation is not consistent with the thermalization property because it neglects the effect of coupling to the bath. In comparison, the Redfield equation correctly takes the MFG state, expanded up to the second order of coupling between the system and bath, as its steady state [14]}. The perturbative evaluation of the steady state of the Redfield equation has some subtleties, since only the second-order coherence (off-diagonal) components can be determined self-consistently [15]}, [16]}, [17]}. In general, correction to the population (diagonal) components in the steady state requires fourth-order expansion of the Redfield equation [15]}, [16]}, [17]}. However, the analytic continuation method used in Ref. [14]} provides a useful way to evaluate the second-order population within the Redfield equation; the result coincides with the second-order coherence and population of the MFG state [14]}.
